Degree Spectra of Differentially Closed Fields Russell Miller - - PowerPoint PPT Presentation

Degree Spectra of Differentially Closed Fields

Russell Miller

Queens College & CUNY Graduate Center

Recursion Theory Seminar University of California – Berkeley 14 April 2014 Joint work with Dave Marker.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 1 / 15

Spectra of Countable Structures

Let S be a structure with domain ω, in a finite language. Definition The Turing degree of S is the join of the Turing degrees of the functions and relations on S. If these are all computable, then S is a computable structure. Definition The spectrum of S is the set of all Turing degrees of copies of S: Spec(S) = {deg(M) : M ∼ = S & dom(M) = ω}. So the spectrum measures the level of complexity intrinsic to the structure S.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 2 / 15

Facts About Spectra

Theorem (Knight 1986) For all countable structures S but the automorphically trivial ones, the spectrum of S is upwards-closed under Turing reducibility. Many interesting spectra can be built using graphs, including upper cones, α-th jump cones {d : d(α) ≥T c}, and more exotic sets of Turing

• degrees. (Greenberg, Montalb´

an, and Slaman recently constructed a graph whose spectrum contains exactly the nonhyperarithmetic degrees.) Indeed, graphs are complete, in the following sense: Theorem (Hirschfeldt-Khoussainov-Shore-Slinko 2002) For every countable structure S in any finite language, there exists a countable graph G which has the same spectrum as S.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 3 / 15

Spectra of Algebraically Closed Fields

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 4 / 15

Spectra of Algebraically Closed Fields

{ all Turing degrees }.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 4 / 15

Differentially Closed Fields

A differential field is a field along with a differential operator δ on the field elements, respecting addition (δ(x + y) = δx + δy) and satisfying the product rule δ(x · y) = (x · δy) + (y · δx). Such a field K is differentially closed if it also satisfies the Blum axioms: for all differential polynomials p, q ∈ K{Y},

• rd(q) < ord(p) =

⇒ (∃x ∈ K) [p(x) = 0 & q(x) = 0], where the order r = ord(p) is the largest derivative δrY used in p. This theory DCF0 is complete and decidable and has quantifier

• elimination. Moreover, it has computable models:

Theorem (Harrington, 1974) For every computable differential field k, there exists a computable model K of DCF0 and a computable embedding g of k into K such that K is a differential closure of the image g(k).

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 5 / 15

Noncomputable Differentially Closed Fields

By analogy to ACF0, one may guess that all countable models of DCF0 have computable presentations. However, it is known that there exist 2ω-many (non-isomorphic) countable models of DCF0. Indeed: Theorem (Marker-M.) For every countable graph G, there exists a countable K | = DCF0 with Spec(K) = {d : d′ can enumerate the edges in some G∗ ∼ = G}.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 6 / 15

Noncomputable Differentially Closed Fields

By analogy to ACF0, one may guess that all countable models of DCF0 have computable presentations. However, it is known that there exist 2ω-many (non-isomorphic) countable models of DCF0. Indeed: Theorem (Marker-M.) For every countable graph G, there exists a countable K | = DCF0 with Spec(K) = {d : d′ can enumerate the edges in some G∗ ∼ = G}. It is not difficult to show that, for every G, there is another graph H s.t. {d : d′ enumerates the edges in some G∗ ∼ = G} = {d : d′ ∈ Spec(H)}, and that conversely, for each H, there is some such G. So the theorem proves that every countable graph H yields a K | = DCF0 with Spec(K) = {d : d′ ∈ Spec(H)}.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 6 / 15

Coding a Graph G into K | = DCF0

Start with a copy ˆ Q of the differential closure of Q. Let A be the following infinite set of indiscernibles in ˆ Q: A = {a0, a1, . . .} = {y ∈ ˆ Q : δy = y3 − y2 & y = 0 & y = 1}. Each am ∈ A will represent the node m from G. Let Eaman be the elliptic curve defined by the equation y2 = x(x − 1)(x − am − an). The coordinates of all solutions to this curve in (ˆ Q)2 are algebraic over Qam + an and Eaman forms an abelian group, with exactly j2 j-torsion points for every j, and with no non-torsion points. There is a homomorphism of differential algebraic groups from Eaman into a vector group, whose kernel E♯

aman is called the Manin kernel of Eaman.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 7 / 15

Coding a Graph G into K | = DCF0

For each m < n with an edge in G from m to n, add a generic point of E♯

am+an to our differential field. The coordinates of this point will each

be transcendental over Qam + an. Let K be the differential closure of the resulting differential field. Thus the coding is: G has an edge from m to n ⇐ ⇒ (∃(x, y) ∈ E♯

aman)[x is transcendental over Qam + an].

In particular, the points we added do not accidentally give rise to any transcendental solutions to any other E♯

am′an′.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 8 / 15

Spec(K) = {d : d′ enumerates some G∗ ∼ = G}

Now if d is the degree of a copy K ∗ ∼ = K, then with a d′-oracle, we enumerate the edges in some G∗ as follows. Find all elements a∗

m of

the set A∗ of indiscernibles in K ∗, go through all solutions to Ea∗

ma∗ n for

each m < n, and ask whether each is transcendental over Qa∗

m, a∗ n

and lies in E♯

a∗

ma∗

• n. If we ever get an answer ”YES,” we enumerate (m, n)

into the edge relation of the graph G∗. Thus G∗ ∼ = G: the isomorphism comes from restricting the isomorphism K ∗ → K to A∗ → A. Conversely, if D ∈ d and D′ enumerates the edges in some G∗ ∼ = G, we build K ∗ ∼ = K using a d-oracle. Start building ˆ Q∗, finitely much at each step. At stage s, if it appears (from D) that D′ has enumerated an edge (m, n) in G∗, add a point xmn ∈ E♯

a∗

ma∗ n which is not (yet) algebraic

• ver Qam, an. If D′ later changes and wipes out this enumeration, we

can still make xmn a t-torsion point for some large t, hence algebraic. Finally, use Harrington’s theorem to build a D-computable differential closure K ∗ of the D-computable differential field defined here.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 9 / 15

Low and Nonlow Degrees

For every d′ > 0′, there exists a graph G such that d′ enumerates a copy of G, but 0′ does not. Therefore: Corollary For every nonlow degree d (i.e., with d′ > 0′), there exists some K | = DCF0 of degree d such that K is not computably presentable. We now prove the converse: Theorem (Marker-M.) Every low model of DCF0 is isomorphic to a computable one. This recalls the famous theorem of Downey-Jockusch that every low Boolean algebra is isomorphic to a computable one.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 10 / 15

Principal Types over k

Over a field E, the principal 1-types are generated by the formulas p(X) = 0, where p ∈ E[X] is irreducible. Over a differential field k, this is not enough! Over Q, the differential polynomial (δY − Y) is irreducible, but only the following formula generates a principal type: δY − Y = 0 & Y = 0. In general, we need pairs (p, q) from k{Y}, with ord(p) > ord(q). If the formula p(Y) = 0 = q(Y) generates a principal type, then (p, q) is a constrained pair, and p is constrainable. Every principal type is generated by a constrained pair, but not all irreducible p(Y) are

• constrainable. p(Y) = δY is a simple counterexample.

Fact p ∈ k{Y} is constrainable ⇐ ⇒ p is the minimal differential polynomial

• f some x in the differential closure K of k.

It is Πk

1 for (p, q) to be constrained, and Σk 2 for p to be constrainable.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 11 / 15

Low Differentially Closed Fields K

If K is low, then the (computable infinitary) Π0

1-theory of K has degree

0′, hence is computably approximable. This allows us to “guess” effectively at the minimal differential polynomial of any x ∈ K over the differential subfield Qxi0, . . . , xin ⊆ K generated by an arbitrary finite tuple from K. Writing K = {x0, x1, . . .} and guessing thus, we build a computable differential field F = {y0, y1, . . .} and finite partial maps hs : K → F such that: (∀n) lims hs(xn) exists; and (∀m) lims h−1

s (ym) exists; and

∀s hs is a partial isomorphism, based on the approximations in K to the minimal differential polynomials of its domain elements. Thus h = lims hs will be an isomorphism from K onto F.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 12 / 15

Differences from Boolean Algebras

The Downey-Jockusch Theorem has been extended. Theorem (Downey-Jockusch; Thurber; Knight-Stob) Every low4 Boolean algebra is isomorphic to a computable one. In contrast, the first Marker-M theorem established that every nonlow Turing degree computes some K | = DCF0 with 0 / ∈ Spec(K). Fact There exists a low Boolean algebra which is not 0′-computably isomorphic to any computable Boolean algebra. (Downey-Jockusch always gives a 0′′-computable isomorphism.) But the theorem for low differentially closed fields built a ∆2 isomorphism onto the computable copy.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 13 / 15

Relativizing the Result

Relativizing the previous theorem yields: Corollary For every K | = DCF0, Spec(K) respects the equivalence relation c ∼1 d defined by c′ = d′. Proof: If c ∈ Spec(K) and d′ = c′, then d can guess effectively at the minimal differential polynomials in the c-computable copy of K, and the process in the theorem builds a d-computable copy of K. Corollary (cf. Andrews, Montalb´ an, unpublished, using Richter) For every K | = DCF0, Spec(K) cannot be contained within any upper cone of Turing degrees, except the cone above 0. Proof: no other upper cone respects ∼1.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 14 / 15

Why Is This a Converse?

Corollary (Marker-M.) For a set S of Turing degrees, TFAE:

1

S is the spectrum of some K | = DCF0.

2

S is the spectrum of some ANT graph and S respects ∼1.

3

S is the preimage under jump of the spectrum of some ANT graph. (ANT: automorphically non-trivial.) (1 = ⇒ 2) was the relativized version of the second theorem (plus the HKSS theorem), and (3 = ⇒ 1) was the first theorem. For (2 = ⇒ 3), if S = Spec(G), let H be the jump of the structure G (defined in work of Montalb´ an and Soskov-Soskova). By HKSS, we may take H to be a

• graph. Then Spec(H) = {c′ : c ∈ Spec(G)}, and so

Spec(G) ⊆ {d : d′ ∈ Spec(H)}. For ⊇, if d′ ∈ Spec(H), then d′ = c′ for some c ∈ Spec(G), and d ∈ Spec(G) since Spec(G) = S respects ∼1.

Russell Miller (CUNY) Spectra of DCF0 Recursion Theory Seminar 15 / 15